Green, Brown, and Probability and Brownian Motion on the Line

This invaluable book consists of two parts. Part I is the second edition of the author's widely acclaimed publication Green, Brown, and Probability, which first appeared in 1995. In this exposition the author reveals, from a historical perspective, the beautiful relations between the Brownian motion process in probability theory and two important aspects of the theory of partial differential equations initiated from the problems in electricity — Green's formula for solving the boundary value problem of Laplace equations and the Newton–Coulomb potential.

Part II of the book comprises lecture notes based on a short course on “Brownian Motion on the Line” which the author has given to graduate students at Stanford University. It emphasizes the methodology of Brownian motion in the relatively simple case of one-dimensional space. Numerous exercises are included.

Sample Chapter(s)
Chapter 1: Some Notation and Terminology (186 KB)

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• Green, Brown, and Probability:
  • Green's Ideas
  • Probability and Potential
  • Process
  • Random Time
  • Markov Property
  • Brownian Construct
  • The Trouble with Boundary
  • Return to Green
  • Strong Markov Property
  • Transience
  • Last but Not Least
  • Least Energy
• Brownian Motion on the Line:
  • Exit and Return
  • Time and Place
  • A General Method
  • Drift
  • Dirichlet and Poisson Problems
  • Feynman–Kac Functionals
• Stopped Feynman–Kac Functional:
  • Introduction
  • The Results
  • The Connections